A lion and its tamer are standing inside a cage that measures 1 unit in radius. Both of them can effortlessly run at the speed of 1 unit maximum.

1) Suppose the lion is hungry, will he be able to eat up the tamer?

2) If yes, how long can the tamer survive?

There can be two possibilities:

The tamer runs on the perimeter

If he runs on the perimeter, the lion will eventually catch him. If the lion follows the radius of the tamer, his trajectory will be half the circle and not a spiral. Thus he will catch the tamer in a finite time.

The tamer does not runs on the perimeter

If he does that, clearly he has an escape strategy and that will work for an infinite time.